# Consider ๐(๐ ) = ๐ 3 + ๐2๐ 2 + ๐1๐  + ๐0 with all real coefficients It is known that its derivative ๐โฒ(๐ ) has no real roots

Q. Consider ๐(๐ ) = ๐ 3 + ๐2๐ 2 + ๐1๐  + ๐0 with all real coefficients. It is known that its derivative ๐โฒ(๐ ) has no real roots. The number of real roots of ๐(๐ ) is

(A) 0ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย  (B) 1ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย  (C) 2ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย  (D) 3

Ans: 1

Sol:

Given p(s) = s3 + a2ย s2 + a1s + a0

p'(s) = 3s2 + 2a2s + a1

Also given that p'(s) has no real roots, therefore, we can write:

n – 1 = 0

n = Number of real roots in p(s)

โด n = 1

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